Question

A certain sum of money, invested at certain rate of Compound Interest, compounded annually, Amounts to Rs.8820 in 2 yrs and Rs. 9261 in 3 yrs. Find the Principal.

• A. 8200
• B. 7800
• C. 7500
• D. 8000

Hint:

For compound interest, the ratio of amounts of two consecutive years gives the growth factor. A\(_{n+1}\)/A\(_{n}\) = (1 + R/100).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We have a sum of money (Principal, P) invested at a certain rate (R) of compound interest. We are given the total amount after 2 years and 3 years. We need to find the initial Principal.
Step 2: Key Formula or Approach:
The formula for the amount (A) in compound interest is A = P(1 + R/100)\(^n\), where P is the principal, R is the rate of interest per annum, and n is the number of years.
The interest for one year on the amount at the end of the previous year can be used to find the rate.
Step 3: Detailed Explanation:
Let the Principal be P and the rate of interest be R% per annum.
Amount after 2 years (A\(_{2}\)) = Rs. 8820
\[ P\left(1 + \frac{R}{100}\right)^2 = 8820 \quad --- (1) \] Amount after 3 years (A\(_{3}\)) = Rs. 9261
\[ P\left(1 + \frac{R}{100}\right)^3 = 9261 \quad --- (2) \] To find the rate R, we can divide equation (2) by equation (1):
\[ \frac{P\left(1 + \frac{R}{100}\right)^3}{P\left(1 + \frac{R}{100}\right)^2} = \frac{9261}{8820} \] \[ 1 + \frac{R}{100} = \frac{9261}{8820} \] The interest for the 3rd year is simply the difference between the amounts:
Interest for 3rd year = A\(_{3}\) - A\(_{2}\) = 9261 - 8820 = Rs. 441.
This interest (Rs. 441) is calculated on the amount at the end of the 2nd year (Rs. 8820).
So, R = \( \frac{\text{Interest}}{\text{Principal for that year}} \) \(\times\) 100
\[ R = \frac{441}{8820} \times 100 \] \[ R = \frac{44100}{8820} = 5% \] Now that we have the rate R = 5%, we can substitute it back into equation (1) to find the Principal P:
\[ P\left(1 + \frac{5}{100}\right)^2 = 8820 \] \[ P\left(1 + \frac{1}{20}\right)^2 = 8820 \] \[ P\left(\frac{21}{20}\right)^2 = 8820 \] \[ P\left(\frac{441}{400}\right) = 8820 \] \[ P = 8820 \times \frac{400}{441} \] Since 8820 / 441 = 20,
\[ P = 20 \times 400 = 8000 \] Step 4: Final Answer:
The Principal amount is Rs. 8000.